The Topology of Higgs Bundle Moduli Spaces

Transcription

1 The Topology of Higgs Bundle Moduli Spaces Peter Beier Gothen This thesis is submitted in partial fulfilment of the requirements for the degree of Doctor of Philosophy at the University of Warwick. The Mathematics Institute, University of Warwick, Coventry. August 1995

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3 To Sofia

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5 Contents Acknowledgements Declaration Summary iii v vii 1 Introduction 1 2 General Facts Higgs Bundles Flat Bundles and Higgs Bundles The Theorem of Donaldson and Corlette Hitchin s Equations and Simpson s Theorem Morse Theory The Morse Function Critical Points and Morse Indices Q-bundles Definitions The Higgs Bundle Associated to a Q-bundle The Moduli Space of Rank 3 Higgs Bundles Statement of the Result Strategy of Proof Critical Points of Type (1, 2) and (2, 1) Critical Points of Type (1,1,1) Calculation of the Morse Indices The Moduli Space of Flat Sp(2n, R)-connections Milnor-Wood Type Inequalities A Proof Using Higgs Bundles The Extremal Case The Components of M Sp(4,R) i

6 ii Statement of the Result Strategy of Proof The Structure of the Critical Points The Spectral Curve The d = 0 case The d = 2g 2 case The 0 < d < 2g 2 case Fibrations

7 Acknowledgements First of all, my gratitude goes to my supervisor, Professor Nigel Hitchin, for teaching me so much and for all his generous help. His enthusiasm made work a pleasure. I am also very grateful to Ib Madsen, who taught me a lot and did much for my interest in mathematics. Thanks to Dietmar Salamon, who handled the paperwork related to the submission and examination of this thesis. Many thanks to Alastair King, for informing me about the material of Section 2.4. And to all the people with whom I had many interesting mathematical discussions; in particular to Sergio Santa Cruz and to Thomas Mielke, who also suggested the use of Maple when calculations became unbearable. Last, but not least, I thank all those people in Århus, Warwick, and Cambridge, who made my time as a student such an enjoyable one. I gratefully acknowledge financial support from the Faculty of Science of the University of Aarhus, the Danish Research Academy, and the Danish Science Research Council. iii

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9 Declaration The work in this thesis is, to the best of my knowledge, original, except where attributed to others. The material of Chapter 3 is a revised version of material which has been submitted for the degree of cand. scient. at the University of Aarhus, Denmark, 1993, and which has been published as: P. B. Gothen, The Betti numbers of the moduli space of rank 3 Higgs bundles, Internat. J. Math. 5 (1994), v

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11 Summary We use the interpretation of the moduli space of flat connections on a Riemann surface in terms of Higgs bundles to study the topology of these spaces. We calculate the Betti numbers of the moduli space of stable Higgs bundles of rank 3 and degree 1 with fixed determinant, which is homeomorphic to the space of representations of a universal central extension of π 1 Σ, in SL(3, C). The calculation is done using Morse-Bott theory. The critical submanifolds are closely related to the moduli spaces of vortex pairs. We study the moduli space of flat reductive Sp(2n, R)-bundles and, using Higgs bundles, we obtain an easy proof of a Milnor-Wood type inequality. Furthermore, we study the number of connected components of the moduli space in the case n = 2. vii

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13 Chapter 1 Introduction Let Σ be a closed surface of genus g 2, and let G be a Lie group. There has been continuing interest in studying the space of representations M = Hom(π 1 Σ, G)/G, where G acts by conjugation. From the point of view of gauge theory, M is the moduli space of flat G-connections modulo gauge equivalence: the flat principal G-bundle associated to a representation ρ: π 1 Σ G is P = Σ G, π 1 Σ where Σ is the universal cover of Σ, and π 1 Σ acts on G through ρ by conjugation. For simplicity, assume that G = SU(n). Then, there is a rank n complex vector bundle E associated to P, and the flat connection on P induces a -operator on E. Over a surface, this is always integrable and so, E acquires the structure of a holomorphic bundle. It turns out that with this holomorphic structure, E is semi-stable, i.e., deg(f ) rk(f ) deg(e) rk(e), for all holomorphic subbundles F E. The number deg(e)/ rk(e) is usually denoted by µ(e) and called the slope of E. When E is an SU(n)-bundle, µ(e) = 0, but the notion of stability is important for holomorphic bundles of arbitrary degree. A bundle E is said to be stable if strict inequality holds above, for all non-zero proper subbundles of E. Finally, E is said to be poly-stable if it is a direct sum of stable bundles. Algebraic geometers have studied the problem of constructing moduli spaces of vector bundles, and it turns out that it is not possible to give the 1

14 2 space of isomorphism classes of all bundles on Σ the structure of a variety. However, Mumford showed that by restricting attention to semi-stable bundles, one can obtain a good moduli space. There is, therefore, a map from M to the moduli space of semi-stable bundles of rank n and degree 0, given by taking a flat connection to the semi-stable holomorphic structure induced by it. The famous theorem of Narasimhan and Seshadri [26] states that this map is a homeomorphism. Higgs bundles were introduced by Hitchin [19] and are of interest for a variety of reasons. One is that they provide the framework for the generalisation of the theorem of Narasimhan and Seshadri to non-compact groups. From the point of view of gauge theory, a Higgs bundle over Σ is a pair (A, Φ), where A is a unitary connection on a C complex vector bundle E and Φ Ω 1,0 (Σ; End(E)), satisfying Hitchin s equations. These are a set of elliptic non-linear differential equations for A and Φ, and one consequence is that Φ is holomorphic with respect to the holomorphic structure on E induced by A. From the holomorphic point of view, a Higgs bundle is a pair (E, Φ), consisting of a holomorphic vector bundle E and a Higgs field Φ H 0 (Σ; End(E) K), where K, as usual, is the canonical bundle on Σ. There is a generalisation of the above notion of stability to Higgs bundles: the condition is the same but it only applies to Φ-invariant subbundles of E. A solution to Hitchin s equations gives rise to a holomorphic Higgs bundle by giving E the holomorphic structure induced by the connection A, and this Higgs bundle turns out to be poly-stable. Moduli spaces of Higgs bundles can be constructed both from the gauge theory point of view (this was done by Hitchin [19]) and from the algebro-geometric point of view (done by Nitsure [27] and Simpson [30]). The analogue of the theorem of Narasimhan and Seshadri, that the moduli space of solutions to Hitchin s equations is isomorphic to the moduli space of poly-stable Higgs bundles, is true. It was proved by Hitchin [19] in the case of bundles of rank 2 and, in greater generality, by Simpson [28]. Hitchin observed that if (A, Φ) is a solution to his equations, the connection A+Φ+Φ is a flat SL(n, C) connection. If B is any SL(n, C)-connection, choosing a Hermitian metric on the bundle allows one to write B = A+Φ+Φ for a unitary connection A and a Higgs field Φ. It is a theorem of Donaldson [13] and, more generally, Corlette [10], that when B is a flat reductive connection, there exists a metric on E such that the associated pair (A, Φ) satisfies Hitchin s equations. Thus, the moduli space of flat reductive SL(n, C)-connections is isomorphic to the moduli space of poly-stable Higgs bundles of rank n and degree 0.

15 Introduction 3 Other vector bundles with extra structure have been studied extensively. One example is that of the vortex pairs of Bradlow [6], which consist of a bundle together with a section. There is a stability condition for these, involving a parameter, and stable pairs correspond to solutions of the vortex equation, which also involves a parameter. The moduli spaces of vortex pairs have been studied carefully by Thaddeus [31]. He shows that the moduli spaces for different values of the parameter are related and, among other things, uses this to calculate their Betti numbers, and to prove the Verlinde formula. A generalisation of the vortex pairs are the triples of Bradlow and García- Prada [9], which consist of a pair of bundles and a map between them. These, and also Higgs bundles, are examples of the Q-bundles of Alastair King. We shall not give the definition here, but refer the reader to Section 2.4 for details. There is a stability condition for Q-bundles, which involves a number of parameters. This is a generalisation of the stability condition for vortex pairs and triples. For certain Q-bundles there is an obvious way of defining an associated Higgs bundle and vice versa. For a special value of the parameters involved in the stability condition for Q-bundles, it is obvious that Higgs bundle stability implies Q-bundle stability. We show that the converse is true for a particularly simple kind of Q-bundle. In this thesis, we shall use the interpretation of the moduli space of flat connections in terms of Higgs bundles to study the topology of some of these spaces. An important ingredient is the theorem about Q-bundles mentioned above. We have two main results. The first is a calculation of the Betti numbers of the moduli space of stable Higgs bundles of rank 3 and degree 1 with fixed determinant. This is the space of representations of a universal central extension of π 1 Σ, in SL(3, C). The reason for considering this space, instead of the space of representations of π 1 Σ, is that it is smooth, which is necessary for the calculation to work. We follow the Morse theory approach of Hitchin s calculation in [19] in the rank 2 case, using the L 2 -norm squared of the Higgs field as a Morse-Bott function. The main new ingredient lies in the determination of the critical submanifolds of the Morse function: this involves Bradlow s vortex pairs mentioned above, and as a result, we see the moduli space of rank 2 vortex pairs, for certain values of the parameter, inside the moduli space of Higgs bundles. The proof of this is an application of the theorem about Q-bundles. In order to carry the calculation through, we use Thaddeus calculation in [31] of the Betti numbers of the vortex moduli spaces. The other result is a study of the number of connected components of the moduli space of flat reductive Sp(4, R)-connections on the surface Σ. Even

16 4 though Sp(4, R) is a real group, there is an interpretation in terms of Higgs bundles (see Section 2.2 for details). The idea to use Higgs bundles to study flat connections for a real group is due to Hitchin [19] and [21]. The idea is again to use Φ 2 as a Morse-Bott function. However, the moduli space is not smooth, so this approach does not work directly. For this reason, we use topological invariants of the Higgs bundles to distinguish subspaces of the moduli space, each of which is a union of connected components. It is a consequence of Uhlenbeck s weak compactness theorem that the function Φ 2 is proper so, if we can show that on each of these subspaces the space of local minima of Φ 2 is connected, it follows that these subspaces are in fact connected components. The most obvious of these topological invariants is the first Chern class, d, of the complex vector bundle obtained by a reduction of structure group to the maximal compact subgroup U(2) Sp(4, R). Denote the corresponding subspace of the moduli space of flat Sp(4, R)-connections by M d. From the Higgs bundle point of view, a well-known Milnor-Wood type inequality can easily be proved; this states that d 2g 2. Furthermore, in the extremal case d = 2g 2, we show that there is an isomorphism between M d and the moduli space of poly-stable Higgs bundles of rank 2 with a nondegenerate quadratic form, where the Higgs field is symmetric with respect to the quadratic form and twisted by the square of the canonical bundle. The main point of the proof is to show that the stability conditions coming from the two different points of view are identical. The proof of this is another application of the theorem about Q-bundles. The Higgs bundles with a quadratic form have structure group O(2, C) and hence, we get the Stiefel-Whitney classes w 1 and w 2 as further topological invariants, when d = 2g 2. The case of w 1 = 0 is relatively simple to treat but, when w 1 0, we need to use the spectral curve introduced by Hitchin in [20], and the mod 2 index theorem of Atiyah-Singer to identify the local minima of the function Φ 2 and show that they form connected subspaces. When d < 2g 2, we conjecture that the subspaces M d are connected but, we are only able to prove this when d = 0. We can, however, show that M d, and this gives a lower bound on the number of connected components of M Sp(4,R). We also discuss two possible approaches to a general proof. This thesis is organised as follows: In Chapter 2, we collect some general facts. Section 2.1 is an introduction to Higgs bundles. In Section 2.2, we describe how to interpret flat bundles in terms of Higgs bundles, and which Higgs bundles correspond to flat Sp(4, R)-bundles. The Morse theory approach to studying the moduli space of Higgs bundles is reviewed in Section 2.3. Section 2.4 is a brief intro-

17 Introduction 5 duction to Q-bundles, and we prove the theorem about Q-bundles mentioned above. Chapter 3 contains the calculation of the Betti numbers of the moduli space of Higgs bundles of rank 3 and degree 1. The result is stated in Section 3.1, while the structure of the argument is explained in Section 3.2. Sections 3.3, 3.4, and 3.5 are taken up by the details of the calculation. In Chapter 4, we study the moduli space of flat reductive Sp(2n, R)-connections from the point of view of Higgs bundles. Section 4.1 contains a proof, using Higgs bundles, of a well-known Milnor-Wood type inequality. In Section 4.2, we are concerned with the main result of the chapter. This is the study of the number of connected components of the moduli space when n = 2. Finally, in Section 4.3, we briefly mention a possible application of the aforementioned result to 4-manifolds, which are fibred over a surface.

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19 Chapter 2 General Facts 2.1 Higgs Bundles Let G c be a complex semisimple Lie group and let G 0 G c be the maximal compact subgroup. Denote the Lie algebras of these groups by g c and g 0, respectively. Thus g c has a compact real structure σ; the +1 eigenspace of σ is g 0, and the 1 eigenspace we denote by g 0 (this is the orthogonal complement of g 0 with respect to the Killing form). Let Σ be closed Riemann surface of genus g 2 and let P be a principal G 0 -bundle on Σ, then we have the adjoint bundle Ad P = P g 0. Ad Denote by P c the G c -bundle corresponding to P, then Ad P c has a real structure induced by σ which, for simplicity, we shall call σ as well. Of course the corresponding real subbundle is just Ad P (in other words Ad P c = Ad P C). From the point of view of gauge theory, a G 0 -Higgs bundle is a pair (A, Φ) consisting of a connection A on P and a (1, 0)-form Φ Ω 1,0 (Σ; Ad P c ) satisfying Hitchin s equations F (A) [Φ, σ(φ)] = 0 (2.1) A Φ = 0. (2.2) Here A is the (0, 1)-part of the covariant derivative d A defined by the connection A. Now, suppose that G 0 = SU(n) and G c = SL(n, C); we thus have an associated vector bundle E = P C n. Let (A, Φ) be a solution to Hitchin s G equations; then A defines a holomorphic structure on E through A and equation (2.2) shows that Φ is a holomorphic section of End 0 (E) K. Here 7

20 8 End 0 denotes the trace-free endomorphisms and, as usual, K is the canonical bundle of Σ. From a holomorphic point of view a Higgs bundle can thus be considered as a pair (E, Φ), where E is a holomorphic vector bundle and Φ H 0 (Σ; End 0 (E) K). Note that, from this point of view, there is no need for the Higgs field to be twisted by the canonical bundle K; the concept makes equally good sense for Φ H 0 (Σ; End(E) L), for any holomorphic linebundle L. Furthermore, in the present description the holomorphic determinant bundle of E is Λ n E = O but, of course, there is no need to specify Λ n E in the definition of a Higgs bundle and, correspondingly, no need to require Φ to be trace-free. Finally, one can define Higgs bundles from a principal bundle point of view: if G c is a complex Lie group, a G c -Higgs bundle is a pair (P c, Φ), where P c is a holomorphic principal G c -bundle and Φ H 0 (Σ; Ad P c L) for some fixed holomorphic linebundle L. When G c GL(n, C), a G c -Higgs bundle gives rise to a Higgs bundle in the vector bundle sense, in general with some extra structure reflecting the definition of G c we shall see an example of this with G c = Sp(2n, C). In this thesis we shall stick mainly to the vector bundle point of view. Let us now describe the notion of stability of Higgs bundles, introduced by Hitchin [19]. The slope of a Higgs bundle is defined to be the slope of the underlying vector bundle: µ(e) = deg(e) rk(e). Furthermore, a subbundle F E is called Φ-invariant if Φ(F ) F K; in other words, (F, Φ) is a sub-higgs bundle of (E, Φ). A Higgs bundle is said to be stable if for any non-trivial proper Φ-invariant subbundle F the inequality µ(f ) < µ(e) (2.3) holds. It is called poly-stable if it is a direct sum of stable Higgs bundles. Finally it is called semi-stable if equality is allowed in (2.3). Next we come to the connection between the gauge theoretic and the holomorphic point of view. Let P be a principal bundle with structure group G 0 = SU(n) and let (A, Φ) be a solution to Hitchin s equations on P. Then a vanishing theorem (see Hitchin [19] and Simpson [28]) states that the corresponding Higgs bundle (E, Φ) is polystable. The theorem of Hitchin and, more generally, Simpson, provides the converse. Theorem 2.1 (Hitchin [19] and Simpson [28]). Let (E, Φ) be a polystable Higgs bundle. There is a unique Hermitian metric on E such that

21 2.1 Higgs Bundles 9 the pair (A, Φ) satisfies Hitchin s equations (2.1) and (2.2). Here, A is the unitary connection determined by the Hermitian metric and the holomorphic structure on E. Our objects of study are various moduli spaces of Higgs bundles. The above theorem allows these to be considered from the point of view of both gauge theory and algebraic geometry. We denote by A the space of connections on P ; this is an affine space modeled on Ω 1 (Σ; Ad P ). We denote by Ω the space of Higgs fields Ω 1,0 (Σ; Ad P c ). Denote the space of solutions to Hitchin s equations by C A Ω and let G = Ω 0 (P Ad G 0 ) be the gauge group. From the point of view of gauge theory the moduli space is then M = C/G. The usual type of arguments, involving Sobolev completions, shows that around irreducible solutions to the equations M has the structure of a smooth manifold. However, in general M will have singularities. For details in the rank 2 case see [19]. From the point of view of algebraic, or holomorphic, geometry, we are considering the space of polystable Higgs bundles (E, Φ) modulo isomorphism. From this point of view, moduli spaces of Higgs bundles have been constructed by Nitsure [27] and in greater generality by Simpson [30]. From Theorem 2.1 it follows that M and the algebraic geometry moduli space M alg are diffeomorphic (actually, the holomorphic structure can also be seen on M and it is then isomorphic to M alg ). From now on, we shall identify the two and denote them by M. We shall need the description of the space of infinitesimal deformations T of a Higgs bundle (P c, Φ) given by Biswas and Ramanan in [5]. This is the first hypercohomology of the complex of sheaves 0 Ω 0 (Σ; Ad P c ) ad(φ) Ω 1,0 (Σ; Ad P c ) 0, where Ω p,q (F ) denotes the sheaf of smooth sections of the bundle of F -valued (p, q)-forms. From this, one easily obtains the 5-term exact sequence, first described by Nitsure [27]: H 0 (Σ; Ad P c ) α H 0 (Σ; Ad P c K) β T where α and δ are induced by the map of sheaves γ H 1 (Σ; Ad P c ) δ H 1 (Σ; Ad P c K), (2.4) ad(φ): O(Ad P c ) O(Ad P c K), (2.5) β maps a variation Φ of the Higgs field to (0, Φ), and γ maps a variation (Ȧ, Φ) to the variation in the bundle A. The Higgs bundle (P c, Φ) corresponds

22 10 to a smooth point of M exactly when α is injective, or, equivalently, δ is surjective, and in this case T is the tangent space to M. The dimension of T can be calculated from the Riemann-Roch formula.

23 2.2 Flat Bundles and Higgs Bundles Flat Bundles and Higgs Bundles In this section we give a summary of the connection between flat bundles and Higgs bundles, in order to establish notation, and to have a convenient reference for some key facts The Theorem of Donaldson and Corlette The theorem of Donaldson [13] and, more generally, Corlette [10] provides a way of choosing a preferred metric on a flat bundle. In this subsection we review their results. Let G be a Lie group, and let ρ: π 1 Σ G be a representation of the fundamental group of the surface in G. This representation corresponds to a principal G-bundle P = Σ G, ρ with a flat connection, which we denote by B. Here Σ is the universal cover of Σ, and π 1 Σ acts on G via Ad ρ. Let H G be a maximal compact subgroup. By a metric on P, we mean a section σ of the bundle P/H = Σ ρ G/H, or, equivalently, a π 1 Σ-equivariant map σ : Σ G/H. To see the reason for this terminology, think of the case G = GL(n, C) and H = U(n). In this case, σ defines a Hermitian metric in the associated vector bundle. Having a metric on P is equivalent to having a reduction of the structure group of P, from G to H. Thus, a metric defines an inclusion of a principal H-bundle i: P H P. The maximal compact subgroup H is characterized as follows: the Killing form, restricted to h, is negative definite and h is maximal with this property. Let h be the complement of h with respect to the Killing form, so that g = h h. The adjoint representation of G splits correspondingly, as the direct sum of Ad h and Ad h. Hence we obtain a splitting of Ad P into the Whitney sum Ad h P Ad h. Note that Ad h P = Ad P H. Thinking of the connection B as an element of Ω 1 (P ; g), we can write i B = A + θ,

24 12 where A Ω 1 (P H ; h) defines a connection in P H, and where θ Ω 1 (P H ; h ) is a tensorial form, and therefore defines an element of Ω 1 (Σ; Ad h P ). The metric σ is called harmonic, when it is an extremum of the energy functional E(σ) = 1 dσ 2. 2 Σ Note, that dσ Ω 1 ( Σ; σ T (G/H)) and that G/H is a Riemannian manifold so, the above integral makes sense. Furthermore, G/H has a Levi-Civita connection, and we denote by d the pull-back of this to σ T (G/H). With this notation, σ is harmonic if and only if d (dσ) = 0. In terms of A and θ defined above, this equation is equivalent to d Aθ = 0, (2.6) and the flatness condition F (B) = 0 is equlivalent to the pair of equations F (A) + 1 [θ, θ] = 0 2 (2.7) d A θ = 0. (2.8) The theorem of Donaldson and Corlette is an existence and uniqueness theorem for harmonic metrics: they will not exist in general but, for certain representations ρ they will. A representation ρ of π 1 (Σ), in a real semi-simple algebraic group G, is said to be reductive, if the Zariski closure of ρ(π 1 (Σ)) is a reductive subgroup of G. Theorem 2.2 (Corlette [10] and Donaldson [13]). Let G be a real semisimple algebraic group, and let ρ be a representation in G of the fundamental group of a surface Σ. If ρ is reductive there exists a unique harmonic metric in the associated principal G-bundle P over Σ. Conversely, a bundle with a harmonic metric will give rise to a reductive representation of the fundamental group Hitchin s Equations and Simpson s Theorem We describe how to reinterpret harmonic metrics in terms of solutions to Hitchin s equations. This, in turn, leads to a holomorphic point of view in terms of Higgs bundles.

25 2.2 Flat Bundles and Higgs Bundles 13 General Theory As before let G c be a complex semisimple Lie group, let G 0 G c be the maximal compact subgroup, and denote the Lie algebras of these groups by g c and g 0, respectively. Thus, g c has a compact real structure σ; the +1 eigenspace of σ is g 0, and the 1 eigenspace we denote by g 0, this being the orthogonal complement of g 0 with respect to the Killing form. Let P c Σ be a G c -bundle with a flat reductive connection B. Apply Theorem 2.2 to get a reduction of the structure group to G 0, and decompose B in a G 0 -connection A and a 1-form θ Ω 1 (Ad g 0 P c ). Denoting the tangent space to Σ by T, θ is a section of a bundle with fibre T g 0. Put V c = T c, g c, C where T c is the complexification of T. There is a real structure on V c, induced from the real structure on T c and the compact real structure σ on g c. By abuse of notation, we shall denote this by σ as well. Denote the real subspace by V and the imaginary subspace by V. Note that V c has an additional complex structure I, linear with respect to i, which comes from the complex structure on T. Denote the ±1 eigenspaces of I by V 1,0 and V 0,1. Observe, that the following identifications hold: V = T g 0 R V = T g 0 R V 1,0 = T (1,0), g c C V 0,1 = T (0,1), g c. C Note also, that the projection onto V, Φ Φ σ(φ), gives an isomorphism V 1,0 = V. Thus, we have an identification Ω 1,0 (Ad P c ) = Ω 1 (Ad g 0 P c ), and consequently θ = Φ σ(φ) for a unique Φ Ω 1,0 (Ad P ). Similarly, we have the usual splitting of the connection A in its (1, 0)- and (0, 1)-parts: d A = A + A. The equations (2.6), (2.7), and (2.8) are equivalent to Hitchin s equations (2.1) and (2.2). In this way, we obtain a poly-stable G c -Higgs bundle from the flat reductive bundle P c.

26 14 Letting G G c be a real form, g c has a real structure τ whose +1 eigenspace is g, the Lie algebra of G. We want to consider flat G-connections from the point of view of Higgs bundles. Let P be a principal G-bundle with a flat connection B and apply the theorem of Donaldson and Corlette as above to obtain a harmonic metric, and hence a reduction to a bundle P H with structure group H. Here H G is the maximal compact subgroup, characterized by the Killing form restricted to h being negative definite, and by being maximal with respect to this property. Letting P c be the G c -bundle obtained from the inclusion G G c, we obtain a G c -Higgs bundle. However, now the connection A will be a connection in P H, and hence define a holomorphic structure in the complexification PH c, and the Higgs field will be a (1, 0)-form with values in Ad h,c P H. Conversely, a G c -Higgs bundle, which is of this particular kind, will give rise to a flat G-bundle. Flat Sp(2n, R)-bundles We shall consider flat Sp(2n, R)-bundles, and describe the Higgs bundles obtained from them via the above procedure. In order to do this, we need a concrete realization of the pull-back diagram of Lie algebras u(n) sp(2n, R) sp(n) sp(2n, C). Of course, the corresponding results hold for the Lie groups. Let V = C n = R 2n be an n-dimensional complex vector space with a Hermitian inner product h. This can be written h = g + iω, where g is a real inner product, and ω is a symplectic form. Put E = V V, thus the complex structure on E is I = ( 0 i i 0 ). We can think of E as a quaternionic vector space, with the complex structure J given by ( ). The matrix ( ) defines a real structure σ on E, and there is an isomorphism γ : End R (V) = End σ (E). Here End σ (E) is the space of endomorphisms which commute with σ. Thus, elements of End σ (E) are real with respect to the real structure on End C (E) defined by σ. This real structure is ( ) ( ) a b d c c d b a and hence, End σ (E) consists of matrices of the form ( a b b a ). The map γ is defined as follows: for A End R (V), write A = a + b, where a and b are

27 2.2 Flat Bundles and Higgs Bundles 15 the complex linear and complex anti-linear parts of A, respectively. Thus a End C (V) and b Hom C ( V, V) = Hom C (V, V). Now define ( ) a b γ(a) =. b a The Hermitian metric h defines an isomorphism V = V and, using this, we define a complex symplectic form on E = V V by Ω ( (v, α), (w, β) ) = α(w) β(v). Hence, Ω is given by the matrix J = ( ), and sp(2n, C) consists of those ( a c d b ) for which d = at, b = b T, and c = c T. Here, and throughout, we use the notation A T exclusively to mean the induced map W V for A: V W. Using g to identify V with Hom R (V, R), we see that A sp(2n, R) End R (V) if and only if it satisfies A T i+ia = 0. Splitting A in complex linear and complex anti-linear parts a and b, it follows that a T = a and b = b T. Hence, the map γ realizes the inclusion sp(2n, R) sp(2n, C). The Hermitian metric h on V defines a Hermitian metric on E, and this realizes u(2n) End C (E) as those endomorphisms, which satisfy A = A, where A denotes the adjoint. Finally, use the fact that Sp(n) is the intersection of U(2n) and Sp(2n, C) to obtain sp(n) sp(2n, C). We need to make one more observation: using the metric g to identify V with Hom R (V, R), we have A T = A ; in other words the transpose of a real matrix corresponds to the conjugate transpose of a complex matrix, under the identification C = R 2. Hence, u(n) End R (V) consists of those complex linear endomorphisms, which satisfy a = a, and therefore γ gives inclusions u(n) sp(n) and u(n) sp(2n, R). Let P be a Sp(2n, R)-bundle with a flat reductive connection B. Applying Theorem 2.2, we obtain a poly-stable Higgs bundle (E, Φ). From the preceding discussion we see that this will be of the form ( ) 0 b E = V V and Φ =, (2.9) c 0 where b = b T and c = c T ; in other words and b H 0 (S 2 V K) c H 0 (S 2 V K).

28 16 Conversely, it follows from Simpson s theorem, that any poly-stable Higgs bundle of the form (2.9) gives rise to a flat reductive Sp(2n, R)-bundle. Thus, the space of flat reductive Sp(2n, R)-connections on Σ is homeomorphic to the moduli space of poly-stable Higgs bundles of the type (2.9). We denote this space by M Sp(2n,R). To conclude, we describe the infinitesimal deformations of a Higgs bundle of the type (2.9). The 5-term exact sequence (2.4) becomes H 0 (Σ; End(V )) α H 0 (Σ; S 2 V K S 2 V K) β T γ H 1 (Σ; End(V )) δ H 1 (Σ; S 2 V K S 2 V K). (2.10) Here, α and δ are induced by the map of sheaves where and α 1 α 2 : O(End(V )) O(S 2 V K) O(S 2 V K), α 1 (ψ) = ψb + bψ T, (2.11) α 2 (ψ) = ψ T c cψ. (2.12)

29 2.3 Morse Theory Morse Theory The Morse Function In our analysis of topological properties of the moduli spaces, we shall follow the approach of Hitchin in [19] and use a Morse function which arises as a moment map for an action of the circle on the moduli space M. Here, we give a short review of the method, for more details see [19]. A Kähler metric on A Ω is defined by g((ψ 1, Φ 1 ), (ψ 2, Φ 2 )) = 2i tr(ψ1ψ 2 + Φ 1 Φ 2), and because it is invariant under the action of the gauge group, it descends to M. The corresponding Kähler form is ω 1 (X, Y ) = g(ix, Y ). Furthermore, S 1 acts on M by (A, Φ) (A, e iθ Φ), preserving g and the symplectic form ω 1. The function (A, Φ) 1 2 Φ 2 is a moment map for the S 1 -action with respect to the symplectic form ω 1. We shall use the negative of this as our Morse function: f(a, Φ) = 1 2 Φ 2. (2.13) Frankel [15, p.5] shows that, in this situation, the function f is a nondegenerate perfect Morse function. Therefore, the Poincaré polynomial of the moduli space M is given by the Morse counting polynomial Σ P t (M) = N t λ N P t (N), (2.14) where the sum is over the critical submanifolds, and λ N is the index of the critical submanifold N, i.e., the dimension of the subbundle of the normal bundle, on which the Hessian of f is negative definite. Thus, our task is to find the critical submanifolds and their indices. Of course, this approach can only be applied directly when M is nonsingular. In Chapter 3, we shall consider the moduli space of Higgs bundles of rank 3 with rank and degree coprime, which is smooth. In Chapter 4, where we are only interested in finding the connected components of the moduli space, we shall use a slight variation of the argument, which also applies when the moduli space has singular points Critical Points and Morse Indices The analysis of the critical submanifolds rests on the following observation. Let m be a fixed point for the circle action, represented by (A, Φ). Then, either Φ = 0, or there is a one-parameter family of gauge transformations

30 18 g(θ) such that g(θ)(a, Φ) = (A, e iθ Φ). In the latter case, let ψ = d dθ g(θ) θ=0 be the infinitesimal gauge transformation generating this family. We then have d A ψ = 0 (2.15) [ψ, Φ] = iφ. (2.16) Equation (2.15) says that ψ is covariant constant so, taking eigenbundles for ψ, we obtain a decomposition of E as a direct sum of holomorphic subbundles: E = j U j, (2.17) where ψ acts on U j by im j for some real numbers m j. Furthermore, when (E, Φ) is stable, it follows from (2.16) that consecutive eigenvalues of ψ differ by i, i.e. m j+1 = m j + 1, and that Φ maps U j to U j+1 K non-trivially. When (E, Φ) is only polystable, it splits as a direct sum of Higgs bundles of this type, all of the same slope. Conversely, any stable Higgs bundle of this form represents a critical point of f (see [21]). Next we shall show how to read off the Morse indices at a critical point from the action of the infinitesimal gauge transformation ψ on the tangent space to M (Hitchin did this in a different way in [21, p. 646f.]). This, in turn, can be easily determined from the exact sequence (2.4). Let X be the vector field on M generating the circle action; then grad(f) = ix which is exactly the condition for f to be the moment map. A point m M is a fixed point of the circle action if and only if X m = 0. In this case, we can define an endomorphism H X of T m M by H X = (X) m, where is any connection on T M. Clearly ih X = H f, the Hessian of f. We shall express the eigenvalues of H X in terms of those of ψ. Let Y be the vector field on C generating the circle action there; then Y is given by Y (A,Φ) = (0, iφ). There is also the vector field on C given by ψ which is Z = ( d A ψ, [ψ, Φ]). Finally, there is the vector field Ỹ = Y Z = (d A ψ, iφ [ψ, Φ]). (2.18)

31 2.3 Morse Theory 19 Note that Ỹ vanishes along the fibre F = π 1 (m) exactly when X vanishes at m, i.e., when m is a fixed point. Thus, Ỹ defines an endomorphism H Ỹ of T (A,Φ) C as above, given by HỸ ( A, Φ) = ( [ψ, A], i Φ [ψ, Φ]). (2.19) Clearly the diagram T (A,Φ) C π HỸ T(A,Φ) C π T m M H X T m M commutes, and therefore we see from (2.19) that, if ψ acts on (Ȧ, Φ) with eigenvalues (im, in) then H X acts on their images under π with eigenvalues ( im, i(1 n)) and, consequently, the eigenvalues of H f are ( m, 1 n).

32 Q-bundles The notion of Q-bundle, due to Alastair King, provides a general framework for considering a large number of the variuos kinds of vector bundles with extra structure, which have been studied in recent years. Higgs bundles, the vortex pairs of Bradlow [6], and the triples of Bradlow and García-Prada [9], are all examples of Q-bundles Definitions A quiver is a directed graph, specified by a set of vertices Q 0 and a set of arrows Q 1, together with head and tail maps h, t: Q 1 Q 0. We shall only consider finite quivers, i.e. quivers for which Q 0 and Q 1 are finite. Definition 2.3. A Q-bundle over a Riemann surface Σ is a collection of holomorphic vector bundles {E i } i Q0 over Σ and a collection of holomorphic maps {φ a : E t(a) E h(a) } a Q1. A twisted Q-bundle is given by in addition specifying a linebundle L a for each arrow a. The maps φ a should then go φ a : E t(a) E h(a) L a. We shall only consider Q-bundles of a particularly simple form: let Q be a quiver with k vertices and exactly one arrow connecting each pair of vertices in each direction. Let I = {1,..., k}. We denote the vertices by {v i } i I and the arrows by {a ij } (i,j) I 2, where a ij is the arrow going from v i to v j. We shall consider Q-bundles where all the maps are twisted by a fixed linebundle L. Thus a Q-bundle, twisted by L, is a pair E = (E, Φ), where E = {E 1,..., E k } and Φ = {φ ij }. Here, each E i is a holomorphic vector bundle on Σ and φ ij is a holomorphic section of Hom(E j, E i L). There is a stability condition for Q-bundles, which generalises the stability conditions for the bundles with extra structure mentioned above. In our case this condition is as follows. It depends on k parameters α = {α 1,..., α k }. The α-slope of a Q-bundle E is by definition µ α (E) = (αi rk(e i ) + deg(e i )). rk(ei ) Note that this only depends on the topological types of the bundles E i. A Q-bundle is α-semi-stable if µ α (F) µ α (E) for any Q-subbundle F. Furthermore, E is α-stable if we have strict inequality above whenever F is non-zero and proper. We shall only have use for the case when all the α i are zero, and

33 2.4 Q-bundles 21 we shall, therefore, use the word stable rather than α-stable. In this case, the α-slope is simply ( ) µ α (E) = µ Ei. (2.20) The Higgs Bundle Associated to a Q-bundle Given a Q-bundle E = (E, Φ), we can define an associated Higgs bundle (E, Φ) by putting E = E i and Φ = (φ ij ), (2.21) i I where (φ ij ) is the matrix of Φ with respect to the above direct sum decomposition of E. Note that (2.20) says that µ α (E) = µ(e). (2.22) Because of this we shall from now on denote the α-slope simply by µ(e), and call it the slope. Let F be a Q-subbundle of E. Then, obviously, F E is a Φ-invariant subbundle. Conversely, if a Φ-invariant subbundle F E is of the form F = F i for subbundles F i E i, then the collection F = {F i } defines a Q-subbundle F of E. From the first of these observations and (2.22), we see that (semi-)stability of (E, Φ) implies (semi-)stability of E. The following theorem says that the converse is true when the Q-bundle E is of a particularly simple form. It would be interesting to investigate whether this holds for more general E. Theorem 2.4. Let Q be a quiver with two vertices and one arrow connecting the vertices in each direction, and let E = ({E 1, E 2 }, {φ 12, φ 21 }) be a Q-bundle. Let (E, Φ) be the associated Higgs bundle as above; thus E = E 1 E 2 and ( ) 0 φ12 Φ =. φ 21 0 Then E is (semi-)stable if and only if (E, Φ) is. Proof. The bundle L by which we are twisting is completely irrelevant to the argument so, for ease of notation, we shall assume that L = O. Also, the following proof is the case of E being semi-stable; we only have to replace by < in (2.25) to obtain the proof in the case of E being stable. Assume that E is semi-stable. We need to show that µ(f ) µ(e) for any Φ-invariant subbundle F E. Let π i : E E i be the projection on the ith factor. Let F i E i and G i F be the subbundles which are generically

34 22 the image and kernel of π i, respectively. Then F 1 and G 2 are contained in E 1, F 2 and G 1 are contained in E 2, and we have sequences of vector bundles 0 G i F F i 0, which are generically short exact. Hence, deg(f ) deg(g i ) + deg(f i ). Putting F = F 1 F 2 and G = G 1 G 2, it follows that and, obviously, 2 deg(f ) deg(f ) + deg(g) (2.23) 2 rk(f ) = rk(f ) + rk(g). (2.24) We claim that F and G are Φ-invariant and, therefore, define Q-subbundles of E. First, let x 1 F 1. If we write x 1 = π 1 (x) for some x = x 1 + x 2 in F, then Φ(x) = Φ(x 1 ) + Φ(x 2 ). By our assumption on the matrix for Φ, it follows that Φ(x 1 ) E 2 and Φ(x 2 ) E 1. Then π 1 (Φ(x)) = Φ(x 2 ) E 1 and π 2 (Φ(x)) = Φ(x 1 ) E 2. But Φ(x) F because F is Φ-invariant, and thus Φ(x 2 ) F 1 and Φ(x 1 ) F 2. Of course, we can repeat the argument with x 2 F 2 and hence, F is Φ-invariant. The proof that G is Φ-invariant is similar. Let x 1 G 2. By assumption, Φ(x 1 ) E 2. But G 2 F, so Φ(x 1 ) F as well. It follows that Φ(x 1 ) G 1 and thus, G is Φ-invariant. We have thus seen that F and G define Q-subbundles of E and from semistability of E, we get deg(f ) rk(f ) rk(g) deg(e) and deg(g) deg(e). (2.25) rk(e) rk(e) Finally, combining (2.23), (2.24), and (2.25), we see that 2 deg(f rk(f ) + rk(g) ) deg(e) rk(e) = 2 rk(f ) rk(e) deg(e). Thus, µ(f ) µ(e) and the proof is finished.

35 Chapter 3 The Moduli Space of Rank 3 Higgs Bundles 3.1 Statement of the Result In this chapter, our aim is to calculate the Betti numbers of the moduli space of rank 3 Higgs bundles on a smooth closed Riemann surface of genus g 2. In order to have a smooth moduli space, we shall restrict attention to bundles which have rank and degree coprime. Thus we let M be the moduli space of stable rank 3 Higgs bundles with fixed determinant bundle Λ 0 of degree d coprime to 3. This space is smooth (see [19]). In this case, the exact sequence (2.4) becomes H 0 (Σ; End 0 (E)) α H 0 (Σ; End 0 (E) K) β T γ H 1 (Σ; End 0 (E)) δ H 1 (Σ; End 0 (E) K). (3.1) The map α is always injective and δ is surjective. From the point of view of gauge theory, we are considering gauge equivalence classes of solutions (A, Φ) to Hitchin s equations (2.1) and (2.2) which, in the present context, take the form F (A) + [Φ, Φ ] = 0 A Φ = 0. Here A is a unitary connection on a rank 3 vector bundle with a fixed induced connection A 0 on the determinant bundle, and Φ is a (0, 1)-form with values in the traceless endomorphisms. F (A) denotes the traceless part of the curvature. 23

36 24 In terms of representations of the fundamental group, we are considering representation of a central extension (cf. [19]): there is a universal central extension Γ of π 1 Σ, generated by elements A 1, B 1,..., A g, B g and a central element C subject to the relation [A i, B i ] = C, and M is the moduli space of irreducible representations of Γ in SL(3, C) which take C to a fixed nontrivial central element determined by the first Chern class of the bundle. Thus, our calculation gives the Betti numbers of this purely topologically defined space. We shall also state the result for the moduli space M where the determinant bundle (of degree d coprime to 3) is allowed to vary. This calculation is actually slightly less involved than the one we shall present. Theorem 3.1. Let Σ be a closed Riemann surface of genus g 2, and let Λ 0 be a holomorphic line bundle on Σ of degree d with (d, 3) = 1. Let M be the moduli space of rank 3 stable Higgs bundles on Σ with fixed determinant bundle Λ 0. The Poincaré polynomial of M is (1 + ( t)4g 4 P t (M) = 2t 2 + t 4 + 2t 2g + 2t 2g+2 1 (1 t) 4 4 t4g 4 3gt 4g 3 + (6g 2 + 2g 3)t 4g 2 + (11g 12g 2 )t 4g 1 + (6g 2 10g )t4g t 8g 6 t 10g 8 ) + t2g (1 + t) 2g 4 ( t 6g 8 (1 + t 3 ) 2g ( 2g t 2 + (2g 2)t 4 ) (1 t) 4 (1 + t 2 ) 2 ) + (1 + t) 2g ( 2t 4 2t 6 + t 2g 4 + 2t 2g 2 + t 2g t 4g 2 ) 22g t 2g (1 + t) 2g 1 + 2gt8g 8 (1 + t) 2g 3 (1 + t 3 ) 2g 1 (1 t) 4 (1 t) 3 (1 + t 2 ) + 22g 1 t 10g 8 (1 + t) 2g (1 t) 3 (1 t 3 ) + t6g 2 (1 + t) 4g 3 (1 + t 2 + t 4 ) (t 1) 3 (1 + t 2 ) 2 (t 6 1) + t 4g 4( (3 2g 1)(1 + t) 4g 4 3 2g). + t4g 4 (1 t) 2g 1 (1 + t) 2g 1 4(1 + t 2 ) + (1 + t5 ) 2g (1 + t 3 ) 2g 1 (t 2 1)(t 4 1) 2 (t 3 1) Let M denote the moduli space of stable Higgs bundles of rank 3 and degree d with (d, 3) = 1 and any determinant. Then, P t (M ) = (1 + t) 2g( P t (M) (3 2g 1)t 4g 4 ((1 + t) 4g 4 1) ). Remark 3.2. It is interesting to note that the Poincaré polynomial of M does not split as the product of those of the Jacobian and M. This is

37 3.1 Statement of the Result 25 in contrast to the case of stable bundles (without Higgs field), see [3]. In particular, it follows that tensoring by a linebundle gives a nontrivial action of the group Γ 3 = {L Jac 0 (Σ) L 3 = O} = (Z/3) 2g on H (M; Q). Remark 3.3. Some simpler results can be obtained from the formulas of Theorem 3.1. Setting t = 1, we see that χ(m) = 3 2g, while χ(m ) = 0. And for a Riemann surface of genus 2, the Poincaré polynomial of M is P t (M) =1 + 3t t t t t t t t t t t t t t t 16.

38 Strategy of Proof The proof of Theorem 3.1 follows Hitchin s Morse theory approach as explained in Section 2.3, using the Morse function f(a, Φ) = 1 2 Φ 2. In this section, we outline how it applies in the case of rank 3 Higgs bundles. The critical points of f corresponding to the absolute minimum f = 0 can be easily dealt with: in this case the Higgs field Φ must vanish and conversely, any Higgs bundle with Φ = 0 is fixed by the circle action. Thus, the corresponding critical submanifold, N 0, is the moduli space of stable bundles of rank 3. The index of N 0 is, of course, λ N0 = 0, and Desale and Ramanan calculated the Poincaré polynomial of N 0 in [11, p.241]. Their formula is P t (N 0 ) =(t 2 1) 1 (t 4 1) 2 (t 6 1) 1( (t 5 + 1) 2g (t 3 + 1) 2g (t 2 + 1) 2 t 4g 2 (1 + t) 2g (1 + t 3 ) 2g + (1 + t 2 + t 4 )t 6g 2 (1 + t) 4g). From Section 2.3, we know that a Higgs bundle representing a critical point is of the form (2.17). When Φ 0, there must be at least two nontrivial summands in the direct sum decomposition of E. Thus, we need to consider three distinct types of critical points: we shall say that a Higgs bundle (E, Φ) (or the critical point it represents) is of type (1, 2) if it is of the form E = L V, (3.2) where rk(l) = 1 and rk(v ) = 2, and where Φ = ( ) 0 0, φ 0 with φ: L V K. Similarly, we say that (E, Φ) is of type (2, 1) if it is of the form E = V L, (3.3) with rk(v ) = 2 and rk(l) = 1, and where Φ = ( ) 0 0, φ 0 with φ: V LK. Note that in both cases stability of (E, Φ) forces φ to be non-zero.

39 3.2 Strategy of Proof 27 with Finally, (E, Φ) is said to be of type (1, 1, 1), if it is of the form E = L 1 L 2 L 3, (3.4) Φ = φ 1 0 0, 0 φ 2 0 where φ 1 : L 1 L 2 K and φ 2 : L 2 L 3 K. Again, φ 1 and φ 2 must be non-zero because (E, Φ) is stable. In Section 3.3, we describe the critical submanifolds corrseponding to Higgs bundles of type (1, 2) and (2, 1), the result being given in propositions 3.7 and 3.9. The critical submanifolds corresponding to Higgs bundles of type (1, 1, 1) are described in Section 3.4, with the result being given in Proposition The indices of the critical submanifolds are given in Proposition From these results, an elementary calculation of the type in [19, pp ] gives the formula of Theorem 3.1. This calculation is rather long, and was done with the help of the computer programme Maple. We omit the details.

40 Critical Points of Type (1, 2) and (2, 1) Let (E, Φ) represent a critical point of f of type (1, 2), hence of the form (3.2). Set d = deg(e), l = deg(l), and v = deg(v ), so that d = l + v. An easy calculation using the Higgs bundle equation F (A) + [Φ, Φ ] = 0 and the fact that F (A) = F (A) F (A 0) I, shows that the (critical) value of f at the point represented by (E, Φ) is f = 2π(l 1 d). (3.5) 3 The fact that (E, Φ) is a stable Higgs bundle, allows us to bound l, and therefore the critical values of f. Proposition 3.4. The degree l of the linebundle L satisfies the inequalities 1 d < l < 1d + g Proof. There is a rank 1 subbundle L V, defined by the requirement that φ: L L K. Thus φ is a non-zero section of the line bundle L 1 L K and hence, deg(l 1 L K) 0. There are three obviously Φ-invariant subbundles of E, namely L, L L, and V. Applying the stability condition to these and combining with the previous inequality gives the stated result. It follows that we can construct any Higgs bundle representing a critical point of type (1, 2) as follows. First, we choose a holomorphic linebundle L of degree l with 1d < l < 1 d + g 1. Then, we choose a rank 2 bundle V 3 3 and a non-zero section φ H 0 (Σ; L 1 V K) such that Λ 0 = Λ 3 (L V ), and set E = L V and Φ = ( ) 0 0 φ 0. But not any V and φ will do; they have to be chosen such that (E, Φ) becomes a stable Higgs bundle. An application of Theorem 2.4 will show that the condition on V and φ is essentially Bradlow s condition of τ-stability (see [6] and [7]). In the case of bundles of rank 2 on a Riemann surface, it takes the following form (cf. Thaddeus [31]). Definition 3.5. Let σ be a positive rational number. A pair (Ṽ, φ) consisting of a holomorphic bundle Ṽ Σ and a non-zero section φ H0 (Σ; Ṽ ) is said to be σ-semi-stable if for any line bundle Ũ Ṽ and deg(ũ) 1 2 deg(ṽ ) σ deg(ũ) 1 2 deg(ṽ ) + σ if φ H0 (Σ; Ũ), if φ H0 (Σ; Ũ). If we have strict inequalities above (Ṽ, φ) is said to be σ-stable.

41 3.3 Critical Points of Type (1, 2) and (2, 1) 29 Let r = deg(ṽ ). For σ 1r (mod Z) and σ < 1 r, smooth moduli 2 2 spaces of σ-stable pairs can be constructed; in [31], Thaddeus constructed a moduli space N (σ, Λ) of pairs with fixed determinant bundle Λ by geometric invariant theory, and in [7], Bradlow and Daskalopoulos constructed a moduli space N (σ) of pairs with any determinant bundle (of degree r). Lemma 3.6. The Higgs bundle (E, Φ) is stable if and only if the pair (Ṽ, φ) is σ-stable for the value σ = 1 6 d l. Proof. Let Q be the quiver with two vertices and just one arrow between them. Then, E = ({L, V }, {φ}) is a Q-bundle twisted by K and (E, Φ) is the Higgs bundle associated to E, as in Section 2.4. The quiver Q satisfies the hypothesis of Theorem 2.4 so, we know that (E, Φ) is stable if and only if the E is stable. But, it is a well known and easy fact that this is equivalent to the pair (Ṽ, φ) being σ-stable for the value of σ given above. For completeness, we shall repeat the argument here. Let L V be defined by φ: L L K. There are two types of non-trivial Q-subbundles of E. The first type is of the form F = ({0, U}, {0}), where U V is a linebundle. Such Q-subbundles are in 1 1 correspondence with linebundles Ũ = L 1 UK Ṽ. The stability condition for these Q-subbundles is deg(u) < 1 3 d, or, equivalently, deg(ũ) < 1 d l + 2g 2. (3.6) 3 The second type of Q-subbundle, of which there is just one example, is F = ({L, L }, {φ}). This Q-subbundle corresponds to the subbundle Ũ Ṽ defined by φ H 0 (Σ; Ũ ). The stability condition for F is deg(l) + deg(l ) < 2 3 d, or, equivalently, deg(ũ ) < 2 d 2l + 2g 2. (3.7) 3 Finally, note that deg(ṽ ) = d 2l + 4g 4. Hence, putting σ = 1d + 1 l, we 6 2 see that the inequality (3.6) corresponds to the second condition in Definition 3.5, while the inequality (3.7) corresponds to the first. We can now determine the critical submanifolds.